
where the penalization term, which represents an object, is expressed with a penalty parameter \(\lambda \) , a mask value \(\chi \) , and a velocity vector of a flow inside the object \({\mathbf {u}}_b\) , which is zero for the fixed object.
The mask value is \(\chi =0\)  in the flow domain and \(\chi =1\)  inside the object.
The size of the computational domain here is \(\left(L_x, L_y, L_z\right)=\left(20D,20D,4D\right)\) .
The computational time step is set to \(\Delta t=5.0\times 10^{-2}\) .
For the training data of PIV example, we focus on the volume around the square cylinder, i.e., \(\left(7D\times 6D\times 0.5D\right)\) .
The number of grid points of the extracted region is \((N_x^*, N_y^*, N_z^*)=(140,120,20)\) .
To consider the three-dimensionality of the present flow at \({Re}_D=300\)  [1]}, twenty \(x-y\)  cross-sections at different spanwise locations are used for the training data.
The details of preparation for particle images can be found in Morimoto et al. [2]}.
